Table 1 Nomenclature of the variables and notation used in the paper
Symbol | Description |
|---|---|
\(\mathcal {V}\) | Set of all agents |
\(\mathcal {L}\) | Set of all locations |
N | Number of agents in the system |
L | Number of locations in the system |
n | Number of agents in each location |
D | Side of the square planar space |
T | Total number of discrete time-steps in the observation window |
t | Index for discrete time instants |
ℓ | Index for the locations in the system |
i,j | Indices for the agents in the system |
ρℓ | Density of location ℓ in the system |
β(i) | Function that maps agent i to its base location |
Ωℓ | Region of space occupied by location ℓ |
xi(t),yi(t) | Position occupied by agent i in the planar space at time t |
θi(t) | Direction of the motion of agent i at time t |
Φi(t) | Angle of the direction of the shortest path from xi(t),yi(t) to Ωβ(i) |
Δθit | Angle drawn uniformly at random in [0,2π) for agent i at time t |
v | Velocity of agents outside their base location |
α | Randomness in the agents’ motion toward their base location |
p | Probability of jumping outside the base location |
Pjump(d) | Probability of jumping at a distance d from the base location |
qin | Probability of being inside the base location |
qout | Probability of being outside the base location |
qℓ | Probability of being inside Ωℓ |
qout,d | Probability of being at a distance d from the closest location |
Ψin,Ψout,Ψd | Probability that agents are inside, outside, and at a distance d |
from their base location | |
P(Σ) | Probability density function of locations’ radii |
γ | Exponents of the power law distribution of locations’ radii |
Σmin,Σmax | Lower and higher cut-off of the distribution of locations’ radii |
G(σ) | Probability density function of agents’ radii of interaction |
ω | Exponents of the power law distribution of radii of interaction |
σmin,σmax | Lower and higher cut-off of the distribution of radii of interaction |
ki | Degree of agent i |
λ | Infection probability per contact |
μ | Recovery probability per unit time |
s(t),i(t),r(t) | Fraction of susceptible, infected, and recovered agents in the system |
sℓ(t),iℓ(t),rℓ(t) | Fraction of susceptible, infected, and recovered agents in Ωℓ |
sout,d(t),iout,d(t),rout,d(t) | Fraction of susceptible, infected, and recovered agents at distance d |
from the closest location | |
Λℓ(t) | Contagion probability in Ωℓ |
Λout,d(t) | Contagion probability at a distance d from the closest location |
〈·〉 | Statistical average of the quantity “ ·” |
E [·] | Expected value of the quantity “ ·” |
P[·] | Probability of an event “ ·” |